The left panel of Fig. REF  shows the results for \(N_t=4\) .
The vertical dashed line represents \(\kappa _c = 0.0602(4)\)  for two-flavor QCD [1]}, which was obtained for infinitely large spatial volume by a finite size scaling analysis including the next-to-leading order effect, i.e., \(n_{\rm max}=N_t+2\) .
The location of the critical point for three-flavor QCD is similar [2]}, [1]}.
From this plot, we see that, around \(\kappa _c\)  of two- and three-flavor QCD, the \(\kappa ^4\)  term is dominant and the effect of the higher-order terms is negligible.
Hence, the determination of \(\kappa _c\)  using the hopping parameter expansion is reliable for \(N_t=4\) .
We also see that the convergence of the hopping parameter expansion suddenly worsens at \(\kappa \mathrel {\unknown. {>}}{\sim }\) 0.13\(, suggesting a lower bound of the convergence radius around there.\) In the right panel of Fig. REF , we show the corresponding results for \(N_t=6\) .
The vertical dashed line shows the critical point \(\kappa _c = 0.1286 (40)\)  for two-flavor QCD obtained by looking at the shape of the histogram of the Polyakov loop on a \(32^3 \times 6\)  lattice [4]}, which was calculated with the leading-order term, i.e., \(n_{\rm max}=N_t\) .
Here, we notice that the critical point determined by the histogram has a large systematic error due to the finite volume effect.
The critical point determined by the histogram is about 10% larger than that determined by a finite size scaling analysis on \(N_t = 4\)  lattices [1]}.
From a full QCD simulation on \(N_t = 6\)  lattices, \(\kappa _c = 0.0877(9)\)  is obtained by a finite size scaling analysis [6]}.
